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\begin{equation*}(ac+bd)^2+(bc-ad)^2 = (a^2 c^2 + b^2 d^2 + 2abcd) + (b^2 c^2 + a^2 d^2 - 2abcd) = a^2 c^2 + b^2 d^2 + b^2 c^2 + a^2 d^2 = a^2(c^2+d^2) + b^2(c^2+d^2) = a^2 + b^2 = 1\end{equation*}
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